Formal Summation of Divergent Series

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 225, 532᎐541Ž. 1998 ARTICLE NO. AY986046

Robert J. MacG. Dawson*

Department of Mathematics and Computing Science, St. Mary’s Uni¨ersity, Halifax, No¨a Scotia, Canada B3H 3C3

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The idea of telescoping a series is widely known, but is not widely trusted. It is often treated as a formalism with no meaning, unless convergence is already established. It is shown here that even for divergent series, the results of telescoping are self-consistent, and consistent with other well-behaved summation operations. Moreover, the summation operations obtained by telescoping are the strongest possible operations with these properties. Some Tauberian theorems are exhibited for telescoping. ᮊ 1998 Academic Press

0. INTRODUCTION

The technique of ‘‘telescoping’’ series is well known, and where applica- ble is very fast and intuitive. A curious and well-known feature of this technique is that it provides a ‘‘sum’’ for many oscillatory and divergent series as well as for many convergent series. Hence, students are often cautioned not to use it without first verifying the convergence of the series analytically. This is to avoid such ‘‘results’’ as Ž.1 q 2 q 4 q 8 q иии y 21Ž.q 2 q 4 q иии s Ž.1 q 0 q 0 q 0 q иии « Ž.1 q 2 q 4 q 8 q иии sy1. Ž 1. This summation, as Bellwx 2 observes, may also be obtained from ‘‘the 1 formal binomial theorem applied toŽ. 1 y 2y .’’ He then goes on to describe it as ‘‘a meaningless result that did not astonish Euler,’’ obtained ‘‘without sufficient attention to convergence and mathematical existence.’’

* Supported by NSERC Grant OGP0046671.

532

0022-247Xr98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. FORMAL SUMMATION 533

In defense of Euler, it should be noted that, although rather alien to real analysis, this equation holds in the field of dyadic numbers, and in the twos-complement arithmetic used by many computerswx 1 . The summation method of Tommwx 8 also sums every power series Ý r i to its formal sum 1r1 y r. It is the purpose of this article to show that telescoping can be put on a rigorous foundation, and its consistency with other methods of summation guaranteed.

1. DEFINITIONS

Let R be a ring with identity 1. A series over R is a vector with elements in R indexed by the natural numbers; it is distinguished from a sequence by the choice of product. Products of sequences are generally defined termwise, reflecting their comparatively weak natural order struc- ture; in contrast, the vector space of series is usually made into an R-algebra using the Cauchy product. This algebra has multiplicative identity 1 s Ž.1 q 0 q 0 q иии . The Cauchy product preserves the right-shift operation ␴ , in the sense that ␴A) B s A)␴ B s ␴ Ž.A) B . The right-shift operation is internally represented by the elementŽ. 0 q 1 q 0 q 0 q иии s ␴ 1 of the algebra of series over R; henceforth, this series is also represented as ␴ . The algebra of series is, of course, isomorphic to the algebra R@␴ # of formal power series over R. We shall sometimes write series as power series; thus the same series may be written asŽ. 1 y 1 q 1 y 1 q иии ,or 23 1 y ␴ q ␴ y ␴ q иии . The subalgebra generated by ␴ is Rwx␴ ,the algebra of polynomials. An element of Rwx␴ may be written omitting terminal 0s; so instead ofŽ. 1 y 2 q 0 q 0 q иии we write 1 y 2␴ . Following axioms A, B, and C of Hardywx 6, p. 6 , we define a summation to be a linear function ᑭ: A ª R, where A is a vector subspace of R@␴ # and is closed under ␴ , ᑭŽ.1 s 1, and ᑭ Ž␴ X .s ᑭ ŽX .. It is not hard to show that any summation takes a finitely supported series to the sum of its nonzero terms. If ᑭ, ᑭЈ are summations, domŽ.ᑭ : dom ŽᑭЈ ., and ᑭЈ NsdomŽᑭ . ᑭ, then we call ᑭЈ an extension of ᑭ and we write ᑭ : ᑭЈ.If ᑭ12and ᑭ agree on the intersection of their domains, then we call them compatible. It is easily shown that: PROPOSITION 1.1. Two summations are compatible if and only if they ha¨e a common extension. For instance, it is known that Euler’s transformation methodw 5, Section 63; 6, Chap. VIIIxw and the Cesaro`¨᎐Holder method 5, Section 60; 6, Chaps. V and XIx are compatible, but each sums some series that the other does not. From the preceding proposition it follows that they have a common proper extension. 534 ROBERT J. MACG. DAWSON

If a summation is compatible with every summationŽ resp., extension of ᑭ.Ž.we call it canonical resp., ᑭ-canonical . For any ᑭ,theᑭ-canonical summations are filtered by the extension relation. Thus, there is a unique maximal extension that extends every canonical extension of ᑭ. This will be called the closure of ᑭ. A series is in the domain of this closure if and only if there is a unique value to which a summation compatible with ᑭ can sum it. Naturally, a summation which is its own closure is called closed. Familiar examples of summations include the finite summation ᑛ, whose domain is the algebra Rwx␴ of series with finite support; the standard summation ᑭ a on the algebra of absolutely convergent series; and the standard summation ᑭ c on the vector space of convergent series. By definitionwx 6, p. 10 a regular method of summation is an extension of ᑭ ca. Note that both ᑭ and ᑭ cmay be factored into the form ⌳⌺ where ⌺ is the linear operator on R@␴ # that takes a series to its sequence of partial sums, and ⌳ takes such a sequence to its limit. ⌺ is represented 1 internally within R@␴ # by the series ⌺ s Ž.Ž.1 q 1 q 1 q иии s 1 y ␴ y . The standard summation of convergent series over the field Zˆ p of p-adic numbers is also a summation of this type. Many other examples of summations, which can sum some divergent series, are given in Hardywx 6 . Some are based on generating functions, such as the methods of Abel and Lindelof.¨ Others, such as the Y method or the Cesaro`¨᎐Holder method, use appropriate linear operators to im- prove summability, applied either to the series itself or to the sequence of partial sums. One important class of summations is the Nørlund means. Let P be a series with terms piP. For any sequence S,let N Ž.S be the sequence whose terms are Ž.Ž.P )S iir⌺ P ;if⌳ N P⌺ Ž.A exists and equals a, we define ᑭ PPŽ.A s a. A Nørlund mean ᑭ is regular iff pirŽ⌺ ŽP ..iª 06, Žw Section 4.1x.Ž . Any two regular Nørlund means are compatible; so by Proposition 1.1, there is a universal Nørlund summation ᑭ N that extends all regular Nørlund means.. If P is ᑭ-summable and does not sum to 0, we may define TP Ž.A to be the series whose terms are Ž.Ž.P ) A rᑭ P .If ᑭ is a limit of partial sums, this is closely related to NP . PROPOSITION 1.2. Suppose that P g domŽ.ᑭ where ᑭ s ⌳⌺, and either ⌳ preser¨es products or P is finitely supported; then ⌳⌺ TPPŽ.A s ⌳ N ⌺ Ž.A Proof. By definition n apjk Ž.⌺ TP Ž.A n s T ÝÝᑭ P is0 jqksi Ž. 1 j s apjk ᑭ P ÝÝ Ž.iqjsnks0 FORMAL SUMMATION 535

1 nnyk s pakj ᑭ P ÝÝ Ž.ks1 js0

⌺Ž.P n s Ž.NP ⌺Ž.A n .2Ž. ᑭŽ.P N

If P is finitely supported, ⌺Ž.P n s ᑭ Ž.P for large enough n.If⌳ preserves products, then

⌺Ž.P ⌳⌺ TPPŽ.A s ⌳ N ⌺Ž.A T ž/ᑭŽ.P N ⌺Ž.P s ⌳⌳Ž.NP ⌺Ž.A ž/ᑭŽ.P N

s ⌳ NP ⌺Ž.A .3Ž.

In the case where P is finitely supported, this is the familiar process of telescoping a series, which we consider in the next section. If P is not finitely supported, but ⌳ preserves products, TP yields a generalization of telescoping which will be considered in a sequel to this article.

2. THE TELESCOPIC EXTENSION

A summation is not required to preserve products; for instance, ᑭ c sums S s Ž.1 y 1r '''2 q 1r 3 y 1r 4 q иии but not S)S. However, even if a summation does not preserve products, linearity and shift-inde- pendence imply a weak multiplicative property.

PROPOSITION 2.1. For any summation ᑭ o¨er R, any element X of domŽ.ᑭ , and any element F of Rwx␴ , F) X g domŽ.ᑭ , and ᑭ ŽF) X .s ᑭŽ.F ᑭ ŽX .. Multiplying a series by an element of Rwx␴ is the same thing as telescoping it: shifting the series to the right one or more times, multiplying the various shifted series by scalars, and adding. This is often done to obtain a series whose sum is known, from which a sum for the original series may be inferred; the next definition makes this rigorous.

DEFINITION. For any summation ᑭ,letTŽ.ᑭ be the summation with domain Ä X: F) X s S, F g Rwx␴ , S g domŽ.ᑭ , ᑭ Ž.F / 04 , such that TŽ.Ž.ᑭ X s ᑭ Ž.S rᑭ Ž.ŽF . Where range Ž.ᑭ is not a field, TŽ.ᑭ takes 536 ROBERT J. MACG. DAWSON values in its rational closure..Ž. We shall call T ᑭ the telescopic extension of ᑭ. The next proposition justifies this notation:

PROPOSITION 2.2. TŽ.ᑭ is well defined, a summation, and is an extension of ᑭ. T preser¨es extensionsŽŽ.Ž.. i.e., if ᑭ : ᑭЈ, then T ᑭ : T ᑭЈ , and TTŽŽᑭ ..s T Žᑭ .. Proof. If a series X is telescoped via two equations F) X s S, FЈ) X s SЈ, we also have FЈ)S s FЈ) F) X s F )SЈ, whence ᑭŽ.S rᑭ ŽF .s ᑭŽ.SЈ rᑭ ŽFЈ .. The proofs of the other assertions follow a similar pattern.

PROPOSITION 2.3. TŽ.ᑭ is the closure of ᑭ. Proof. If ᑭ has an extension ᑭЈ, TŽ.ᑭ and ᑭЈ have Ž by Proposition 2.2.Ž.Ž. a common extension T ᑭЈ and are compatible. Thus T ᑭ is ᑭ-canonical. Let X be a series not in domŽŽT ᑭ .., and let F) X q S s FЈ) X q SЈ for some F, FЈ g Rwx␴ , SЈ, S g domŽ.ᑭ . Then ŽF y FЈ .) X s Ž.SЈ y S ;as X f dom ŽŽ..Ž.Ž.T ᑭ , ᑭ FЈ y F s ᑭ SЈ y S s 0. Thus, the following two summations on Ä F ) X q S: F g Rwx␴ , S g domŽ.ᑭ 4 are well defined:

ᑭ1 : Ž.Ž.F) X q S ¬ ᑭ S , Ž.4 ᑭ 2 : Ž.Ž.Ž.F) X q S ¬ ᑭ F q ᑭ S .

Then ᑭ12Ž.X s 0, ᑭ Ž.X s 1; but ᑭ 1, ᑭ 2are both extensions of ᑭ. We conclude that no summation defined on X can be ᑭ-canonical, and thus if ᑭЈ is ᑭ-canonical, domŽ.ᑭЈ : dom ŽŽ..T ᑭ . Because T Ž.ᑭ is ᑭ-canonical, it is compatible with ᑭЈ; thus TŽ.ᑭ is an extension of ᑭЈ.

COROLLARY 2.3.1. TŽ.ᑛ is compatible with e¨ery summation; and any other summation with this property is a restriction of TŽ.ᑛ . It remains to classify the series which are not TŽ.ᑭ -summable. Clearly, there are three possibilities. It is possible that the only case in which F) X s S, F g Rwx␴ , S g domŽ.ᑭ is that in which F s S s 0; such a series X is called untelescopable over ᑭ. If there exist F, S such that F) X s S, ᑭŽ.F s 0, ᑭ Ž.S / 0, then X cannot have a sum in any extension of ᑭ; these series are called T-infinite over ᑭ, and the set of all such series is represented by IT Ž.ᑭ . It is noteworthy that Eulerwx 7 considered series, such as 1 q 2 q 3 q 4 q иии , which are T-infinite over ᑛ to sum to infinity.

PROPOSITION 2.4. i. A finitely supported series sums to 0 if and only if it is di¨isible, within Rwx␴ , by 1 y ␴ . FORMAL SUMMATION 537

n ii. A series is T-infinite o¨er ᑭ if and only if it is of the form ⌺ ) A where A is ᑭ-summable and ᑭŽ.A / 0.

iii. If ᑭ : ᑭЈ, then ITTŽ.ᑭ : I ŽᑭЈ .. Finally, it is possible that there exist F / 0, S / 0 such that F) X s S, but that in every such case ᑭŽ.F s ᑭ Ž.S s 0. In this case, X will be called ᑭ-pseudo-telescopable. It is not hard to see that a series A is ᑭ-pseudo-telescopable if and only if there exists i G 0 such thatŽ 1 y i i 1 ␴ . ) A f domŽŽT ᑭ .., although T ᑭŽ.1 y ␴ q ) A s 0. We will repre- sent the set of ᑭ-pseudo-telescopable series by ⌿T Ž.ᑭ . These series may be thought of as ‘‘formally telescopable but effectively untelescopable.’’ If X is untelescopable or pseudo-telescopable over ᑭ, then for any element s of any ring S = R there exists an extension ᑭЈ of ᑭ such that ᑭЈŽ.X s s. For instance, let dom ŽᑭЈ .s Ä F) X q S: F g Rwx␴ , S g domŽ.ᑭ 4; and define ᑭЈŽ.Ž.Ž.F) X q S s ᑭ Fsq ᑭ S . The next proposition gathers together easily verifiable algebraic properties of domŽŽT ᑭ ..,

ITTŽ.ᑭ , and ⌿ Ž.ᑭ .

PROPOSITION 2.5. i. domŽŽ..T ᑭ j ⌿TT Ž.ᑭ is a ring, and ⌿ Ž.ᑭ is an ideal in it;

ii. domŽŽT ᑭ ..j ITT Žᑭ .is a ring, and I Žᑭ .is an ideal in it;

iii. domŽŽ..T ᑭ j ⌿TT Ž.ᑭ j I Ž.ᑭ is a ring. Pseudo-telescopable series do not exist for every summation. By Propo- sition 2.4, if F, G are finite series with ᑛŽ.F s ᑛ Ž.G s 0, F and G have a common factor 1 y ␴ that may be divided out, until eventually either n n FrŽ.1 y ␴ or GrŽ.1 y ␴ Ž.or both has a nonzero sum. Thus, no series is pseudo-telescopable over ᑛ. On the other hand, it is shown in the 11 following section that the harmonic seriesŽ. 1 qqq23иии is pseudo- telescopable over the absolutely convergent series.

3. EXAMPLES

ⅷ The series W s Ž.1 y 1 q 1 y 1 q 1 q иии , whose nth term is n Ž.y1 , is an element of domŽŽT ᑛ ..,as1 Žq ␴ .)W s 1. T Žᑛ .Ž.W s 1r2; this value is also given by many classical summation methods. This series was known to Leibnitz, Euler, and James Bernoulliwx 2 , who all assigned this value to it. n ⅷ The series X s Ž.1 q 2 q 4 q 8 q иии , whose nth term is 2 , is an element of domŽŽT ᑛ ..; and T Žᑛ .Ž.X sy1. Euler obtained this value by formal application of the binomial series, and the same value is obtained by Tomm’s method of summationwx 8 ; but most classical methods of summation fail to sum X. 538 ROBERT J. MACG. DAWSON

ⅷ ⌺ s Ž.Ž.1 q 1 q 1 q иии satisfies 1 y ␴ )⌺ s 1, and this is thus T-infinite over ᑛ Ž.hence over all other summations. 11 ⅷ yn The series Y s Ž.1 qqq24иии , whose nth term is 2 , is diver- ˆ 1 gent in Z2 . Telescoping with the multiplierŽ. 1 y 2 yields the sum 2. ⅷ The series Z s Ž.0 q 1 y 1 q 0 q 1 q 0 q 0 q 0 y 1 q иии , whose n n Ž2thtermis.Ž.y1 and whose other terms are 0, is untelescopable over 1 ᑭ ca, and hence over ᑭ and ᑛ. However, Abel’s method sums it to2 . 11 ⅷ The series E s Ž.1 q 1 qqq26иии , whose nth term is 1rn!, is untelescopable over ᑛ Ž.although it is summable by ᑭ a. To prove this, we note that its terms are rational numbers; thus, if it were in TŽ.q , its sum would be rational in any extension of ᑛ. 111 ⅷ The harmonic series H s Ž.1 qqqq234иии , whose nth term is 1rn, is pseudo-telescopable over ᑭ a. It is easily verified that H)Ž.1 y ␴ is absolutely convergent to 0; we see in the following text that H is not properly telescopable over ᑭ a. ⅷ There is no obvious generalization of most classical methods of summation to finite fields, as topologies on finite sets yield only trivial convergence. However, TŽ.ᑛ assigns values Ž possibly infinite . to all peri- odic series over finite fields. For instance, in Z3, it may be verified that TŽ.Žᑛ 1 q 2 q 1 q 2 q иии .s 2, althoughŽ 1 y 2 q 1 y 2 q иии .is T-infinite.

ⅷ Inwx 4 , a generalized face number is defined for regular tesselations of hyperbolic planes. We may ‘‘formally count the faces’’ by dividing the tesselation into concentric shells one face deep, with ni faces in shell i. The series Ž.n01q n q иии is divergent but may always be telescoped over ᑛ: the value obtained is consistent with that via a formal use of Euler’s formula. For instance, in the hyperbolic tesselationÄ4 4, 5 , successive shells starting at a single initial face containŽ. 1, 12, 48, 180, 672, 2508, . . . faces. This may be telescoped asŽ.Ž. 1 q 8 q 1 r 1 y 4 q 1 and thus is evaluated by TŽ.ᑛ to y5.

4. TAUBERIAN AND RELATED THEOREMS

A ‘‘Tauberian theorem’’ is one that limits the domain of a summation method by showing that if a series is summable by that method, and its terms obey some bounding condition, then the series is in fact summable by some weaker method. In this section, we look at some Tauberian theorems for the telescopic extensions of some classical summation methods, and other related theorems. The following simple proposition, for instance, shows that the series which are telescopable over finite addition FORMAL SUMMATION 539 either have exponentially growing terms or are summable by classical methods.

PROPOSITION 4.1ŽŽ.. Tauberian Theorem for T ᑛ . If a series X is in n domŽT Žᑛ .., and Xn s O ŽŽ1 q ⑀ . . for all ⑀ ) 0, then X is Holder¨ summable and Xnns OŽ.1.If X s o Ž.1,then X g domŽ.ᑭ a. Proof. Any such X is a linear combination of geometric series b i 2 n Ý aiiin␴ Ž1 q r q r q иии .Ž.If X s O Ž.1 q ⑀ . for all ⑀ ) 0, then <

Abel’s summation operation, ᑭ A, takes a series with generating function f to the value lim fzŽ.if this existsŽ see 6, Chap. IV .. It was the z 1 wx subject of the original Tauberian theorem; the next theorem is a Taube- rian theorem for TŽ.ᑭ A . n THEOREM 4.1. If a series X is in domŽT Žᑭ An .., then X s OŽŽ1 q ⑀ . . for all ⑀ ) 0 iff X g domŽ.ᑭ A .

Proof. It is knownwx 5, Section 283 that, for any X g domŽ.ᑭ A whatso- n ever, Xn s OŽŽ1 q ⑀ . .Žfor all ⑀ ) 0. Suppose X g dom TŽᑭ A..; then F) X s A where F is finitely supported and A g domŽŽT ᑭ A ... Assume also that X s ArF is in ‘‘lowest terms’’ so that we cannot write FЈ) X s AЈ for a strictly lower degree FЈ and Abel-summable AЈ. A power series is Abel summable at precisely the nonsingular points on n its closed disc of convergence. Suppose Xn s OŽŽ1 q ⑀ . . for all ⑀ ) 0; we show that FzŽ.has no zeros on the open unit disc. Suppose, for a contradiction, that there was such a zero, with absolute value r - 1. Then n n the radius of convergence of Ý Xznnwould be at most r,soÝ Xr is not n n OŽŽ1 y ⑀ . . for any ⑀ ) 0. In particular, it is not OŽŽŽ1 q r .r2;so . . Xn is 1 n not OŽŽŽ1 q ry ..r2. , contradicting our assumption. Furthermore, if X is telescopable, ᑭ AŽ.F / 0, so Fz Ž.does not have a zero at z s 1. There- fore, 1 is a nonsingular point on the closed disc of convergence of X and

X g domŽ.ᑭ A .

THEOREM 4.2ŽŽ Tauberian Theorem for T ᑭ c... If a series X is in domŽŽT ᑭ cn .., and X s o Ž.1,then X g domŽ.ᑭ c.

Proof. Any X g domŽŽT ᑭ c .. may be written as ArF s Ar ŽŽ␴ y ␣1 .иии Ž␴ y ␣mi .. where A is convergent and ␣ / 1 for all i. Ži.Žiy1. Ži. Let XЈ s X )Ž.␴ y ␣1 , and X s X )Ž.␴ y ␣in. Then <Ž X . < s < m Ž Žiy1.. < m <Ž Žiy1.. < Ž. Ý js0 XFnyjjF Ý js0 X nyjkmax F ; so if the terms of X con- Ž m. vergeŽ. as a sequence to 0, so do those of XЈ, XЉ,...,X s A.Itis therefore sufficient to show that if X )Ž.␴ y ␣ s A, with A convergent 540 ROBERT J. MACG. DAWSON and Xn s oŽ.1 , then X is convergent; the more general result follows by induction. Ž ␣ ␣ 2 иии . nn␣ yi X s A) 1 q q q ;so Xnis Ý s0 Ai . The nth partial sum of X is given by

Ž.⌺ X n s Ý Xi is0 ni jyi s ÝÝAi ␣ is0 js0

nnyi j s ÝÝAi ␣ is0 js0

n i 1 n 1 y ␣ y q s Ai Ý 1 ␣ is0 y 1 nn nyiq1 s ÝÝAiiy A ␣ 1 y ␣ ž/i 0 i 0 s s 1 ␣ s Ž.⌺ A n y Xn ,5Ž. 1 y ␣ 1 y ␣ which is, by hypothesis, convergent. Exactly the same argument yields:

THEOREM 4.3ŽŽ.. Tauberian Theorem for T ᑭ a . If a series X is in domŽŽT ᑭ an .., and X s o Ž.1,then X g domŽ.ᑭ a.

Note. The seriesŽ. 1 y 1 q 1 y иии , which is telescopable over ᑭ a, and hence over ᑭ c, but summable by neither, shows that these last two theorems are as strong as possible. The harmonic series was stated, in the previous section, to be pseudo- telescopable over ᑭ a. In fact, it follows from the three preceding Taube- rian theorems that it is pseudo-telescopable over ᑭ ac, ᑭ , and ᑭ A. This may be generalized:

THEOREM 4.4. If the termsŽ. Xn of a series are o Ž.1 and X is not conërgentŽ. resp., not absolutely conërgent , then X is pseudo-telescopable oër ᑭ caŽ.resp., ᑭ .

Proof. The proof for ᑭ cawill be given; that for ᑭ is almost identical. The series X )Ž1 y ␴ .ŽŽ.Žs X01021q X y X .Žq X y X .q иии .has Xn as its nth partial sum, so that if Ž.Xncs o Ž.Ž1,ᑭ X ) Ž1 y ␴ ..s 0. Thus, FORMAL SUMMATION 541

X must be either telescopable, T-infinite, or pseudo-telescopable over ᑭ c. By Theorem 4.2, X cannot be telescopable over ᑭ c. Moreover, X cannot satisfy any equation of the form X )Ž.1 y ␴ s A, where T ŽŽ..ᑭ c A / 0, because this would imply X s A)⌺, making X the series of partial sums of A. Either A g domŽ.ᑭ c , in which case its partial sums converge to ᑭ cŽ.ŽA assumed nonzero. and are not oŽ.1;or A is telescopableŽ or again T-infinite.Ž but nonconvergent, and again its partial sums cannot be o 1.. Thus A is not T-infinite; so it must be pseudo-telescopable.

COROLLARY 4.4.1. All conditionally conërgent series are pseudo-telescopable oër ᑭ a. 2 THEOREM 4.5. For any power series PŽ. z s Žp01q pzq pz 2q иии ., PŽ. z is T Žᑭ a .-summable or T-infinite oër the entire complex plane if and only if PŽ. z is the Taylor series of a meromorphic function without a pole at 0. Proof. If the generating function has a meromorphic extension, there are only finitely many poles Ä4␣ii, each of finite order n , in any disc of the form <

ACKNOWLEDGMENT

I thank the various colleagues who have been supportive of this work, especially R. Wood and E. Heighton.

REFERENCES

1. M. Beeler, R. W. Gosper, and R. Schroeppel, ‘‘HAKMEM,’’ MIT Artificial Intelligence Memo No. 239, MIT, Cambridge, MA, 1972. 2. E. T. Bell, ‘‘The Development of Mathematics,’’ McGraw-Hill, New York, 1945. 3. R. P. Boas, ‘‘Invitation to Complex Analysis,’’ Random House, New York, 1987. 4. R. J. MacG. Dawson, A generalized face number for hyperbolic honeycombs, in prepara- tion. 5. K. Knopp, ‘‘Theory and Application of Infinite Series,’’ Blackie, London, 1959. 6. G. H. Hardy, ‘‘Divergent Series,’’ Clarendon, Oxford, U.K., 1949. 7. M. Kline, Euler and infinite series, Math. Mag. 56 Ž.1983 , 307᎐314. 8. L. Tomm, A regular summability method which sums the geometric series to its proper value in the whole complex plane, Canad. Math. Bull. 26 Ž.1983 , 171᎐180.